Astm c 970 87 (2006)

Designation C 970 – 87 (Reapproved 2006) Standard Practice for Sampling Special Nuclear Materials in Multi Container Lots1 This standard is issued under the fixed designation C 970; the number immedia[.]

Designation: C 970 – 87 (Reapproved 2006)

Standard Practice for

Sampling Special Nuclear Materials in Multi-Container

This standard is issued under the fixed designation C 970; the number immediately following the designation indicates the year of

original adoption or, in the case of revision, the year of last revision A number in parentheses indicates the year of last reapproval A

superscript epsilon (e) indicates an editorial change since the last revision or reapproval.

1 Scope

1.1 This practice provides an aid in designing a sampling

and analysis plan for the purpose of minimizing random error

in the measurement of the amount of nuclear material in a lot

consisting of several containers The problem addressed is the

selection of the number of containers to be sampled, the

number of samples to be taken from each sampled container,

and the number of aliquot analyses to be performed on each

sample

1.2 This practice provides examples for application as well

as the necessary development for understanding the statistics

involved The uniqueness of most situations does not allow

presentation of step-by-step procedures for designing sampling

plans It is recommended that a statistician experienced in

materials sampling be consulted when developing such plans

1.3 The values stated in SI units are to be regarded as the

standard

1.4 This standard does not purport to address all of the

safety problems, if any, associated with its use It is the

responsibility of the user of this standard to establish

appro-priate safety and health practices and determine the

applica-bility of regulatory limitations prior to use.

2 Referenced Documents

2.1 ASTM Standards:2

E 300 Practice for Sampling Industrial Chemicals

2.2 Other Standard:

NUREG/CR-0087, Considerations for Sampling Nuclear

Materials for SNM Accounting Measurements3

3 Terminology Definitions

3.1 analysis of variance—the body of statistical theory,

methods, and practice in which the variation in a set of

measurements, as measured by the sum of squares of the measurements, is partitioned into several component sums of squares, each attributable to some meaningful cause (source of variation)

3.2 confidence interval—(a) an interval estimator used to

bound the value of a population parameter and to which a

measure of confidence can be associated, and (b) the interval

estimate, based on a realization of a sample drawn from the

population of interest, that bounds the value of a population parameter [with at least a stated confidence]

3.3 Estimation, Estimator, Estimate:

3.3.1 Estimation, in statistics, has a specific meaning,

con-siderably different from the common interpretation of guess-ing, playing a hunch, or grabbing out of the air Instead, estimation is the process of following certain statistical prin-ciples to derive an approximation (estimate) to the unknown value of a population parameter This estimate is based on the information available in a sample drawn from the population

3.4 estimator—a function of a sample (X 1 , X 2 , , X n) used

to estimate a population parameter

N OTE 1—An estimator is a random variable; therefore, not every

realization (x 1 , x 2 , , x n ) of the sample (X 1 , X 2 , , X n) will lead to the same value (realization) of the estimator An estimator can be a function that, when evaluated, results in a single value or results in an interval or

region of values In the former case the estimator is called a point estimator, and in the latter case it is referred to as an interval estimator.

3.5 estimate, (a: n)—a particular value or values realized by

applying an estimator to a particular realization of a sample,

that is, to a particular set of sample values (x 1 , x 2 , , x n ) (b:

v)—to use an estimator.

3.6 nested design— one of a particular class of experimental

designs, characterized by “nesting” of the sources of variation:

for each sampled value of a variable A, a given number of values of a second variable B is sampled; for each of these, a given number of values of the next variable C is sampled, etc.

The result is that each line of the “Expected Value of Mean Square” column in an analysis of variance table contains all but one of the terms of the preceding line

3.7 random variable— a variable that takes on any one of

the values in its range according to a [fixed] probability distribution (Synonyms: chance variable, stochastic variable, variate.)

1

This practice is under the jurisdiction of ASTM Committee C26 on Nuclear

Fuel Cycle and is the direct responsibility of Subcommittee C26.08 on Quality

Assurance and Reference Materials.

Current edition approved Jan 1, 2006 Published February 2006 Originally

approved in 1982 Last previous edition approved in 1997 as C 970 – 87 (1997).

2 For referenced ASTM standards, visit the ASTM website, www.astm.org, or

contact ASTM Customer Service at service@astm.org For Annual Book of ASTM

Standards volume information, refer to the standard’s Document Summary page on

the ASTM website.

3

Available from National Technical Information Service, Springfield, VA 22161.

Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.

3.8 standard deviation (s.d.)—the positive square root of the

variance

3.9 variance—(a: population) the expected value of the

square of the difference between a random variable and its own

expected value; that is, the second moment about the mean (b:

sample) The sum of squared deviations from the sample mean

divided by one less than the number of values involved

4 Significance and Use

4.1 Plans for sampling and analysis of nuclear material are

designed with two purposes in mind: the first is related to

material accountability and the second to material

specifica-tions

4.2 For the accounting of special nuclear material, sampling

and analysis plans should be established to determine the

quantity of special nuclear material held in inventory, shipped

between buyers and sellers, or discarded Likewise, material

specification requires the determination of the quantity of

nuclear material present Inevitably there is uncertainty

asso-ciated with such measurements This practice presents a tool

for developing sampling plans that control the random error

component of this uncertainty

4.3 Precision and accuracy statements are highly desirable,

if not required, to qualify measurement methods This practice

relates to“ precision” that is generally a statement on the

random error component of uncertainty

5 Designing the Sampling Plan—Measuring Random

Error

5.1 The random error component of measurement

uncer-tainty is due to the various random errors involved in each

operation such as weighing, sampling, and analysis The

quantification of the random error is usually given in terms of

the variance of the mean of the measurements When analyzing

a lot of nuclear material to estimate the true concentration, p,

of a constituent such as uranium, the sample mean, p¯, is the

calculated estimator The variance of p¯, s p ¯2, is a measure of

the random error associated with the measurement process

This practice deals primarily with random error; measurement

process systematic error will be discussed briefly in8.2

5.2 To estimate the true concentration, p, in a lot consisting

of N containers using a completely balanced nested design,

randomly select n of the N containers; from each of the n

containers, randomly select m samples; perform r laboratory

analyses on each of the nm samples (It is assumed that the

amount of material withdrawn for samples is only a small

fraction of the total quantity of material.) Let

X ijk 5 measured concentration of the constituent in the k th analysis

on the jth sample from the i th container, or

where:

p = true concentration,

b i = effect due to container i,

s ij = effect due to the j th sample from container i, and

a ijk = effect due to the k th analysis on the j thsample from

container i.

Then, if each container holds the same amount of material,

(Note 2), the sample mean

p¯ 5 X ¯ 5 1

nmr (

i 5 1

n

(

j 5 1

m

(

k 5 1

r

is an estimator of the true value p The true variance of p¯ is

then

sp¯ 5sb

n

~N 2 n!

N 2 1 1

ss2

nm1

sa

where:

sb = true variance among the N containers in the

given lot, defined as N −1 (p i 2 − N −2 ((p i ) 2;

ss 2 = true variance among samples taken from a

single container,

sa 2 = true variance of the laboratory analysis on

a homogeneous sample, and

N 2 n

N 2 1

= finite population correction factor

N OTE 2—If the ith container has g igrams of material, then the true average concentration is (1N

w ip i , where w i = g i/(1N

gi However, the

variance of the corresponding estimate can still be calculated as shown in

this guideline; the true variance will be only slightly larger if the g ivalues

do not differ too much For example, if the s.d of the g iwere 20 % of the

average g i , it can be shown that the s.d of p would be underestimated by about 2 % of the true standard deviation; for g i’s having s.d.’s of 10 % or

30 % of their average, the underestimation is 0.5 % or 4.5 % respectively.

Note that a set of 25 weights g i, uniformly spread from 3.3 to 6.7 kg, has

a s.d equal to 20 % of the average (5 kg) (It is assumed that errors in the estimation of net weights are insignificant compared to differences between containers, sampling variability, and analytical uncertainty, or both.)

5.3 Since the true variances sb, ss2, and sa are generally unknown, they may be estimated using appropriate data Those data can be historical data obtained from analyzing production samples, as long as there have been no changes in the process with time If such data are not available, as for example during the start-up of a facility or after a change in process conditions,

a designed experiment is required to obtain estimates of the variances.4

5.4 An estimate s p ¯2of the variance of the sample mean can

be obtained from Eq 3, by inserting estimates of the variances appearing there If a designed experiment is performed, the estimates can be obtained from the mean squares

It is shown inAppendix X1 that estimates of the variances are as follows:

s s25 1

r ~MS s 2 MS a!, (5)

s b 5N 2 1 Nmr ~MS b 2 MS s!, (6)

where:

MS a, MSb , and MS s are the “mean squares” for analyses,

4

This topic can be found in many standard statistical texts, for example,

Brownlee, K A., Statistical Theory and Methodology in Science and Engineering,

2nd ed., John Wiley and Sons, New York, 1965; Bennett, C A., and Franklin, N L.,

Statistical Analysis in Chemistry and the Chemical Industry, John Wiley and Sons,

New York, 1954; Mendenhall, William, Introduction to Linear Models and the

Design and Analysis of Experiments, Duxbury Press, Belmont, CA, 1968; and in

Jaech, J L., “Statistical Methods in Nuclear Material Control,” (TID-26298, USAEC, 1973).

containers and samples The estimated variance of p¯ is

ob-tained by replacing the true variances in Eq 3 by their

estimates:

s ¯ p25 1

n

N 2 n

N 2 1 s b 1 1

nm s s

2 1 1

nmr s a (7)

Finally, expressed in terms of the mean squares, this

be-comes

s p¯ 5 1

nmr

N 2 n

N MS b1 1

Nmr MS s. (8)

5.5 The variance of the sample mean, sp ¯2, or its estimate, s

p ¯2, is used to calculate confidence limits for the quantity and

concentration of nuclear materials Therefore, it is desirable to

reduce this variance and, in this way, reduce the random error

Obviously, this can be done by using large values of n, m, and

r (large number of samples and laboratory analyses) The cost

and time required by that approach could be prohibitive

Another approach is to improve the overall process such that

the basic variances sb2, ss2, sa2are reduced

5.6 Eq 8 gives an estimate of the variancesp ¯2for any given

n, m, and r and therefore can be used for comparing different

sampling plans An example of two sampling plans involving

the same number of analyses but having different random

errors is given inAppendix X3

5.7 When one has fixed resources within which the

sam-pling plan must function, the question arises as how to allocate

these resources to obtain the “best” sampling plan Sections6

and7discuss this problem when “cost” is considered “Cost”

is used generically here—it need not be a monetary quantity; it

could be time or something else

6 Determining Sample Sizes

6.1 There are two common situations in which sampling

plans must be developed for use in nuclear material

measure-ment when there are constraints on resources In the first

situation a constraint is imposed upon the “cost” of sampling

and analysis In this case, the problem is to find a plan that

minimizes the variance of the sample mean (minimizes random

error) subject to the cost constraint In the second situation, a

constraint is imposed upon the variance of the sample mean

(upon the random error) and the problem is to find a plan which

minimizes cost subject to this constraint Since this latter

problem is the most frequently encountered, methods for its

solution will be given The former problem, for which the

solution technique closely parallels the one given, will be

covered in footnotes

6.2 Component Variances Are Known:

6.2.1 If the variance constraint is expressed as a maximum

value for the width, 2D, of a confidence interval for p, it can be

transformed immediately to a maximum value for sp ¯, by using

the relationship

D 5 ~Z12a/2!s¯ p

(9)

where:

Z 1-a/2 = value having a probability a/2 of being exceeded by a

standard normal variate

Therefore, if D is limited to Do, say, then sp ¯is limited to Do/

Z 1−a/2 Since the minimum cost is achieved when the constraint

is barely satisfied, we need to minimize cost subject to the constraint

where K is a constant, either specified directly or computed

from Doand a

6.2.2 When the underlying variances are known from pre-vious history, the problem of achieving a minimum cost within

a stated confidence interval width reduces to finding a suitable

set of values for n, m, and r InAppendix X2it is shown that

the optimum r and m are given by

r 5ssa

sSc s

c aD1 / 2

(11)

m 5sss

bSc b

c s

N 2 1

N D1 / 2

(12)

where:

c b = marginal cost of choosing one additional container and preparing it for sampling,

c s = marginal cost of drawing an additional sample from a container and preparing it for analysis, and

c a = marginal cost of an additional laboratory analysis

Therefore, the optimum values for r and m do not depend on

n, and in fact can be calculated immediately from the

vari-ances, the “costs,” and N.

6.2.3 Once m and r are determined and inserted into Eq 3, s

p ¯2is seen to be a monotonic decreasing function of n, so that one need only make n large enough to achieve the required

bound on s p ¯2(Note 3) Letting c s= ca= cb= 1.0 provides the

optimum values of r, m, and n when costs are considered equal.

In practice, the optimum values for m and r obtained this way

are unlikely to be integers Unless these values are very close

to integers, it is prudent to consider both bracketing values, that

is, if the optimum value for r is1.4, try both r = 1 and r = 2 The reason is that the final value of n will generally be different and it is not clear beforehand which set of values of r, m, and

n will achieve the required variance at minimum cost It is also

possible to use different values of m (or r, or both) for different

containers or samples, or both, to obtain a non-integer

“effec-tive” value of m (or r, or both) In this case, p¯ should be

replaced by a weighted average; s p ¯2becomes more compli-cated; and the expected values of the mean squares also become more complicated, as does the estimate of s p ¯2 The advice of a statistician is strongly suggested if this approach is being considered

N OTE 3—The same values of m and r provide minimum variance for

given cost When these are inserted into the cost function, it is seen to be

proportional to n, so that n should be chosen as large as the cost constraint

will allow.

6.2.4 An example with further discussion is given in Ap-pendix X3

6.3 Component Variances Are Not Known:

6.3.1 The approach to finding values for n, m, and r

described in Appendix X2 is also valid when the basic variances are not known, provided some estimates of these variances are available As in 6.2, values for m and r can be

C 970 – 87 (2006)

obtained from estimates of the variances and cost factors.

There is a complication in the calculation of an optimum value

of n, however: since the final uncertainty will be based not on

the true variances but rather on estimates, the t-distribution4

must be used instead of the Normal Given the allowable

half-width D, we have

where:

t1−a/2(n) = value having probability a/2 of being exceeded by a

“Student’s t” variable with degrees of freedom n, and s

p ¯= estimated standard deviation of the mean

Unfortunately, n depends upon n, m, and r (and if prior data are

to be combined with present data in computing s p ¯it depends

also upon the degrees of freedom appropriate to those data)

We therefore proceed iteratively We guess n, calculate n (as

described in6.3.2), obtain t from standard tables, and calculate

s p ¯from Eq 13 We then use this target value and our estimates

of the basic variances to obtain an optimum value for n as in

6.2.3 If this optimum value is as large as, but not too much

larger than, the guessed value, it should be used Otherwise,

use it in place of the initial guess and repeat the procedure

6.3.2 The uncertainty in the final p¯ will be expressed in

terms of an estimated variance s p ¯2 The t-factor used with s p ¯

in Eq 13 has been shown to be approximately correct, provided

the degrees of freedom parameter, n, is properly chosen

Satterthwaite’s formula is applicable, whether or not the data

from a prior experiment are to be used In the simple case

where only the n 3 m 3 r data values under consideration are

used, the formula5is

n 5 s ¯ p4F N 2 n

nmrND2

MS b

n 2 11S 1

mrND2

MS s2

n~m 2 1!G21

(14)

When prior data are combined with these data, the formula

is more complicated

6.3.3 When n and m are both greater than one, the approach

given here leads to an unbiased estimate of sp ¯2 If n or m, or

both, are chosen to be one, then the corresponding mean

square(s) (Appendix X1) are undefined If n = 1, no estimate of

sp ¯2is available If n > 1 and m = 1, then only an overestimate

of s p ¯ 2is available: (1/nmr) MS b has expected value (s b2/n)

(N/(N − 1)) + (s s2/nm) + (s a2/nmr), in which the first term is

too big by the factor (N/(N − 1)) Therefore, in order to avoid

this problem, it is desirable to choose n greater than one; and

unless N is large, also choose m greater than one.

7 Compositing Samples

7.1 In the example ofAppendix X3 at least seven samples

and seven laboratory analyses (measurements) were needed to

reduce the variance of the sample mean to the specified value

Laboratory measurements are usually costly and time

consum-ing Sampling operations, on the other hand, are relatively

inexpensive from the viewpoint of required instrumentation

and operator time Furthermore, in many SNM accountability

situations the variance components due to between- and

within-container variabilities are not known with the same degree of confidence as the laboratory variance To reduce the effort in the laboratory and to minimize the random error, it could be desirable to blend samples to form a composite

7.2 When each container in a lot (n = N) is sampled m times with r analyses per sample, the finite population correction

factor in Eq 3 becomes zero and Eq 3 becomes:

sp¯ 5 1

N Sss2

m 1

sa

mrD5 1

Nm Sss21sa

r D (15)

If the m samples from each individual container are

com-posited and thoroughly mixed (Note 4) and each of the N composites is analyzed r times, Eq 15 is replaced by:

sp¯ 5 1

NSss2

m 1

sa

The laboratory effort is still rather large, since even for r = 1

a total of Nr = N measurements must be made.

N OTE 4—Thorough mixing is very important to give effective homog-enizing of the composite samples, thereby reducing the error from subsampling to a negligibly small value.

7.3 To further reduce the laboratory effort, the m samples from each of the N containers in the lot may be composited into

a lot master sample and thoroughly mixed The contributions to

the master sample from each of the N containers should be

proportional to the net weights in the corresponding containers

A sub-sample (Note 5) of the composite is then analyzed r

times The variance of the sample mean is given by

sp¯ 5ss

2

Nm1

sa

N OTE 5—Dissolution of the material is a step in the laboratory analysis; therefore, the sub-sample must contain an amount of material sufficient for

further subdivision into r portions.

7.4 For this latter case, it is shown inAppendix X4that the

values of m and r that minimize “cost” for a given variance bound k are

m 5sN sS=c asa1=c sss

r 5 s aS=c asa1=c sss

Finally, the minimum cost is given by

minimum cost 51

k ~=c asa1=c sss!2 (20)

(Note 6)

Note that, while Eq 18 and Eq 19 give m and r, the values

will not generally be integers If the values are rounded to integers, then Eq 20 is not appropriate for calculating the actual

cost corresponding to the chosen m and r Instead, the cost would be calculated as c s Nm + c a r.

N OTE 6—If it is desired to minimize the variance for given cost C, the

same technique leads to

mN=c s

r=c a

sa 5

C

=c sss1=c asa , and (21)

5Mendenhall (op cit), p 352; also Jaech (op cit), pp 157–161.

the minimum variance is given by

minimum variance 5C1 ~=c asa1=c sss!2 (22)

7.5 An example with further discussion is given in

Appen-dix X5

7.6 Compositing in this way has a major drawback, in that

it is impossible to estimate ss2, the within-container variance,

on a continuing basis Quite possibly ss 22 may change,

especially if there has been a change in process conditions or

supplier Periodically, and especially at those times when a

change in ss 2might be expected, a number of samples may be

drawn from each container and analyzed separately in replicate

to establish current estimates of ss2and sa

8 Mechanical and Physical Aspects of Sampling

8.1 The common types of nuclear material encountered are

liquid and solids (powder and pellets) In whatever form

encountered, the principal task in sampling is to remove a

sample that is typical of the bulk material, at least as far as the

parameters of interest are concerned The selection of a

procedure and equipment for sampling must be made based on

factors such as the following:

8.1.1 Type and form of the material,

8.1.2 Degree of homogeneity,

8.1.3 Stability of the material,

8.1.4 Location of the material,

8.1.5 Purpose and requirements for analyzing the material,

and

8.1.6 Accessibility for sampling all units or containers

involved

8.1.7 Practice E 300 and NUREG/CR-0087 present

prin-ciples and guidelines for sampling materials The mechanical

and physical aspects of sampling are discussed

NUREG/CR-0087 addresses sampling nuclear materials to determine their chemical and isotopic contents

8.2 Some Sources of Error:

8.2.1 There are various sources of error in the sampling process such as nonhomogeneity, contamination of the sample after removal from the bulk material, failure of the equipment, failure of the operator to follow the procedure, bias, and chemical and physical changes in the material during sampling The latter two sources of error are discussed briefly in8.2.2and 8.2.3

8.2.2 Bias occurs when, in addition to the random errors, all measured values are shifted consistently from the true value in the same direction Likely sources of bias are improper sampling procedures, faulty sampling devices, and improperly calibrated instruments The problem is to detect the existence

of such biases and to account for them in the results The solution to this problem usually requires designing an appro-priate study.6This may not always be possible For example, if the sample were contaminated to an unknown degree and new samples are not available, it may be impossible to estimate the bias

8.2.3 An example of errors due to chemical or physical changes occurs with plutonium dioxide powder.7This material, which is usually handled as a fine powder with a large surface area, readily picks up or loses water if exposed to a change in humidity Plutonium dioxide powder can gain over 1 % in weight within a few hours if exposed to an increase in humidity Therefore, very careful control over conditions must

be established and maintained when sampling this material, particularly if it is relocated and then sampled

APPENDIXES (Nonmandatory Information) X1 ESTIMATION OF VARIANCES IN A NESTED ANALYSIS OF VARIANCE DESIGN

X1.1 Let X ijk be the kth measurement on the jth sample

from the ith container, k = 1, , r; j = 1, , m; i = 1, , n.

Let

X ij.5 (

k 5 1

r

X ijk , X ¯ ij.5 1

X i 5 (

j 5 1

m

(

k 5 1

r

X ijk , X ¯ i 5 1

mr X i , and (X1.2)

X 5 (

i 5 1

n

(

j 5 1

m

(

k 5 1

r

X ijk , X ¯ 5 1

X1.1.1 Then the mean squares may appear in a nested analysis of variance (ANOVA) table as follows:

Source Mean Square Expected Value of

Mean Square Containers MS b = [mr/(n − 1)]

· ((n

i = 1 (X ¯ i − X ¯ )2

[mrN/(N − 1)]s b

+ rs s

2

+ sa

Expected Value of Mean Square Samples MS s = [r/n(m − 1)]

·(n

i = 1(m

j = 1 (X ¯ ij. − X ¯ i )2

rs s2+ sa

Analyses MS a = [1/nm(r − 1)]

·(n

i = 1(m

j = 1(r

k = 1 (X ijk − X ¯ ij.) 2

sa

6

Stephens, F B., et al, Methods for the Accountability of Uranium Dioxide,

NUREG-75/010, pp 1–17, U S Nuclear Regulatory Commission, National Technical Information Service, Springfield, VA, 1975.

7Gutmacher, R G., et al, Methods for the Accountability of Plutonium Dioxide,

USAEC Report WASH-1335, 1974.

C 970 – 87 (2006)

Note that factor [(N/(N − 1)] is due to the finite number of

containers From the preceding table, it is seen that estimates of

the variances are as follows:

s s25 1

s b 5N 2 1

X1.1.2 In practice the latter two estimates could be negative which would require modification of this estimation procedure.4

X2 FINDING THE OPTIMAL VALUES OF r AND m FOR MINIMIZING “COST” SUBJECT TO THE

CONSTRAINT THAT s p¯2 = K (see6.2 )

X2.1 The total cost8of sampling and analysis is not linear

(in n, m, and r) over the whole range of these variables.

However, in the neighborhood of the optimum, a linear

approximation is likely to be reasonable Write the variable

part of the cost as

c 5 c b n 1 c s m8 1 c a r8 (X2.1)

where:

m8 = mn,

r8 = nmr and, as in6.2.2,

c b = marginal cost of choosing one additional container

and preparing it for sampling,

c s = marginal cost of drawing an additional sample from a

container and preparing it for analysis, and

c a = marginal cost of an additional laboratory analysis

Then applying the Lagrange multiplier technique,9we

con-sider the expression

L 5 C 1 l~s p¯ 2 K!, (X2.2)

where (see Eq 3, S 5.2 ):

sp¯ 5 sb

n ·

N 2 n

N 2 11

ss2

m8 1

sa

Taking partial derivatives with respect to r8, m8, n, and l,

setting them equal to zero, and solving for l gives

l 5 c a

sa r825 c s

ss2m825c b

sb

N 2 1

N n

2

(X2.4)

From this it follows that the optimum r and m are given by

r 5 m8 r8 5ssa

sSc s

c aD1 / 2

m 5 m8 n 5ss

sbSc b

c s

N 2 1

N D1 / 2

(X2.6)

X3 EXAMPLE

X3.1 Find values of n, m, and r that meet a variance

constraint and minimize “cost” (6.2.4) Let N = 20; s b= 0.3;

ss= 0.1; and s a = 0.04 For a = 0.05, Z 1−a/2= 1.96 and if

D= 0.2, the target value for sp ¯2is given by:

K 5 D2/Z12a/2 5 0.22

From Eq 3 in X2, r 5 0.040.1 c s

1/2

c a ,

which is close to 1 whenever 5 <cs⁄ca< 10 Since the cost of

sampling is unlikely to be more than ten times the cost of an

analysis and since r $ 1, r will usually be taken equal to one.

Likewise, m 5 0.10.3~c b/c s 3 19/20!1/2 , which is close

to 1 whenever 8 <cb⁄cs< 15 Since m $ 1, m will usually be

taken to be one With r = m = 1,

sp¯ 5 0.09

n

20 2 n

0.01

n 1

0.0016

5 0.10633

setting sp ¯2= 0.0104, we obtain n = 7.03.

Thus the required variance is (approximately) achieved by

taking n = 7, m = r = 1.

X3.2 The previous paragraph shows that at least 7 contain-ers out of the 20 must be selected, sampled once, and each sample analyzed once to obtain the target value of 0.0104 for

sp ¯2and therefore meet the specified confidence interval width (2D) of 0.4 The seven containers must be selected at random, that is, each of the 20 containers is assigned one of a sequence

of numbers and a random number table is used to select the seven containers

X3.3 Note that in this case, the variance term involving

variance bound of 0.0104, unless n $ 7; and the other terms contribute so little that for n = 7, the total variance is down to the required value even for m = r = 1 Therefore, (a) we need

n $ 7 and (b) once n = 7, m and r do not need to be any larger

than their minimum value, so these optimum values are really independent of the costs This will not always be so, of course X3.4 Significant improvements in the variance s p ¯ 2 can

8 Cost need not be monetary.

9

Mendenhall, op cit, p 355.

sometimes be achieved with small additional cost by

judi-ciously choosing values for n, m,and r This is apparent by

comparing the sets: n = 7, m = 2, r = 1 and n = 14, m = 1,

r = 1 In both situations 14 sampling operations and 14

analyses are required, that is, the total effort is about equal

However, for the second set ( n = 14), the variance is 0.0029, which is lower by a factor of three than for the first set (n = 7), which has a variance of 0.0096 Therefore, the set with n = 14

is preferred, unless the cost of choosing additional containers is quite large

X4 FINDING THE OPTIMAL VALUES FOR r AND m—COMPOSITE SAMPLE CASE (7.4)

X4.1 Selection of m and r to minimize cost for a given

variance bound K is achieved by the Lagrange multiplier

technique that was used inAppendix X2 The function to be

considered is

L 5 c s Nm 1 c a r 1 lSss2

Nm1

sa

Setting the partial derivatives with respect to m and r equal

to zero, and solving for l yields

l 5c s N2m2

ss2 5

c a r2

sa

Then, usingss2⁄Nm +sa2⁄r = K , the values for m and r can

be given in symmetric form as

mN=c s

r=c a

sa 5

=c asa1=c sss

X5 EXAMPLE

X5.1 Find values of m and r that meet a variance constraint

and minimize “cost”—composite sample case (7.5)

X5.1.1 A lot consists of N = 20 containers.

Let

ss 2 = 0.01

sa 2 = 0.0025

c s = 1 unit

c a = 16 units

X5.1.1.1 Suppose the variance of p¯ is not to exceed

k = 0.001 Then, from X4,

mN

0.15

4r

0.055

0.2 1 0.1

so that

m 5300 3 0.1

X5.1.1.2 The minimum cost is found to be 90 units If m and

r are rounded up to 2 and 4, the actual cost is, of course,

1 3 20 3 2 + 16·4 = 104 units, and the actual variance is 0.000875 If instead of compositing, one sample were taken from each container and analyzed, the cost would be 340 units, and the variance would be 0.000625

ASTM International takes no position respecting the validity of any patent rights asserted in connection with any item mentioned

in this standard Users of this standard are expressly advised that determination of the validity of any such patent rights, and the risk

of infringement of such rights, are entirely their own responsibility.

This standard is subject to revision at any time by the responsible technical committee and must be reviewed every five years and

if not revised, either reapproved or withdrawn Your comments are invited either for revision of this standard or for additional standards

and should be addressed to ASTM International Headquarters Your comments will receive careful consideration at a meeting of the

responsible technical committee, which you may attend If you feel that your comments have not received a fair hearing you should

make your views known to the ASTM Committee on Standards, at the address shown below.

This standard is copyrighted by ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959,

United States Individual reprints (single or multiple copies) of this standard may be obtained by contacting ASTM at the above

address or at 610-832-9585 (phone), 610-832-9555 (fax), or service@astm.org (e-mail); or through the ASTM website

(www.astm.org).

C 970 – 87 (2006)